# Question: Racetracks give the odds for each horse winning a race

Racetracks give the odds for each horse winning a race rather than the probability. Here are the odds for the 2011 Belmont Stakes.
Odds at a racetrack indicate the amount won for a bet on the winning horse. For example, these racetrack odds mean that a \$1 bet on Master of Hounds wins a \$5 payout (plus the \$1 wager) if Master of Hounds wins the race.
True odds, rather than racetrack odds, are equivalent to probabilities. If you know the true odds, you can get the probabilities and vice versa. The true odds for an event E are the ratio of the probability that this event happens to the probability that it does not happen.
(odds (E) = P (E) / 1–P (E)
If you know the odds, you can get the probability from this formula by solving for P(E):
P (E) = odds (E) / 1 + odds (e)
The raecetrack odds shown are the odds against a horse winning.
If these racetrack odds are true odds, then the following calculation illustrated for Master of Hounds shows how to convert them into probabilities:
P (Master of Hounds wins) = 1 / 5+ 1 = 1 /6
A bet is known as a fair bet if the chance of winning equals the share of the payout. For example, suppose two people each wager a \$1 on the toss of a coin, heads or tails. If the winner gets the total pot of \$2, then the bet is fair. Now suppose that the two bettors wager on the roll of a die, with the first bettor winning if the number is 1 or 2. For the winner-take-all bet to be fair, if the first bettor puts \$2 in the pot, then second bettor must put in \$4.
Motivation
(a) Operating a racetrack for horses is expensive. If these bets are fair, do you think that the track will be able to stay in business?
Method
(b) Suppose that the true odds for Master of Hounds to win the race are 6-1. Why would the racetrack prefer to offer gamblers the odds shown in the table (5-1)?
(c) If the racetrack payouts for these horses are smaller than the payouts implied by the true odds, what will be the sum of the probabilities obtained by treating racetrack odds as true odds?
Mechanics
(d) Treat the racetrack odds as true odds and find the probability for each horse winning the 2011 Belmont Stakes.
(e) What is the sum of the probabilities found in (d)?
Message
(f) Interpret the results of this analysis. Are racetrack odds true odds?

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